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Curves, dual to each other; see below for properties. In projective geometry, a dual curve of a given plane curve C is a curve in the dual projective plane consisting of the set of lines tangent to C. There is a map from a curve to its dual, sending each point to the point
These sets can be used to define a plane dual structure. Interchange the role of "points" and "lines" in C = (P, L, I) to obtain the dual structure. C ∗ = (L, P, I ∗), where I ∗ is the converse relation of I. C ∗ is also a projective plane, called the dual plane of C. If C and C ∗ are isomorphic, then C is called self-dual.
A curve in this context is defined by a non-degenerate algebraic equation in the complex projective plane.Lines in this plane correspond to points in the dual projective plane and the lines tangent to a given algebraic curve C correspond to points in an algebraic curve C * called the dual curve.
These points form a nonsingular quartic curve that has genus three and that has twenty-eight real bitangents. [3] Like the examples of Plücker and of Blum and Guinand, the Trott curve has four separated ovals, the maximum number for a curve of degree four, and hence is an M-curve. The four ovals can be grouped into six different pairs of ovals ...
In mathematics, a duality, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of A is B, then the dual of B is A.
For example, Desargues' theorem is self-dual in this sense under the standard duality in projective geometry. In mathematical contexts, duality has numerous meanings. [ 1 ] It has been described as "a very pervasive and important concept in (modern) mathematics" [ 2 ] and "an important general theme that has manifestations in almost every area ...
The cube and regular octahedron are dual graphs of each other. According to Steinitz's theorem, every polyhedral graph (the graph formed by the vertices and edges of a three-dimensional convex polyhedron) must be planar and 3-vertex-connected, and every 3-vertex-connected planar graph comes from a convex polyhedron in this way.
A plane projective curve is the zero locus of an irreducible homogeneous polynomial in three indeterminates. The projective line P 1 is an example of a projective curve; it can be viewed as the curve in the projective plane P 2 = {[x, y, z]} defined by x = 0. For another example, first consider the affine cubic curve